Stiff Fluid in Accelerated Universes with Torsion

Hindawi Publishing Corporation Journal of Gravity Volume 2013, Article ID 306417, 10 pages http://dx.doi.org/10.1155/2013/306417 Research Article Stiff Fluid in Accelerated Universes with Torsion Almaz Galiakhmetov Department of Physics, Donetsk National Technical University, Kirova Street 51, Gorlovka 84646, Ukraine Correspondence should be addressed to Almaz Galiakhmetov; [email protected] Received 16 February 2013; Accepted 18 March 2013 Academic Editor: Sergei Odintsov Copyright © 2013 Almaz Galiakhmetov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The flat Friedmann universes filled by stiff fluid and a nonminimallyupled co material scalar field with polynomial potentials of the fourth degree are considered in the framework of the Einstein-Cartan theory. Exact general solution is obtained for arbitrary positive values of the coupling constant . A comparative analysis of the cosmological models with and without stiff fluid is carried out. Some effects of stiff fluid are elucidated. It is shown that singular models with a de Sitter asymptotic and with the power-law 4/3 ( ) asymptotic at late times are possible. It is found that =3/8is a specific value of the coupling constant. It is demonstrated that the bouncing models without the particle horizon and with an accelerated expansion by a de Sitter law of an evolution at late times are admissible. 1. Introduction has grown in connection with the fact that torsion arises naturally in the supergravity [33–35], Kaluza-Klein [36–38], Recent cosmic observations [1–6]favoranisotropicspatially and syperstring [39–41]theories.() gravity with torsion flat Universe, which is at present expanding with acceleration. has been developed [42–45] as one of the simplest extensions The source of this expansion is an unknown substance with of the ECT. In [43] it has been demonstrated that, in () negative pressure called dark energy (DE). Establishment of gravity, torsion can be a geometric source for the accelerated the origin of DE has become an important problem. Different expansion. The equivalence between a Brans-Dicke theory theoretical models of DE have been put forward (see, e.g., with the Brans-Dicke parameter 0 =−3/2and the Palatini the reviews [7–10] and references therein). Among these () models various modifications of general relativity (GR) were gravity with torsion has been demonstrated in [44]. () considered, the Einstein-Cartan theory (ECT) in particular gravity has been constructed [46–48] as the extension [11–13]. This theory [14–17] is an extension of GR to a of the “teleparallel” equivalent of GR [49], which uses the space time with torsion, and it reduces to GR when the Weitzenbock¨ connection that has no curvature but only torsion vanishes. The ECT is the simplest version of the torsion. Poincare´ gauge theory of gravity (PGTG). It should be noted At present, the considerably increased precision of mea- that the ECT contains a nondynamic torsion, because its surements in the modern observational cosmology stipu- gravitational action is proportional to the curvature scalar lated its essential progress. In this connection, the exact of the Riemann-Cartan space-time. In this sense, the ECT is cosmological solutions, which makes it possible to elucidate a degenerate gauge theory [17–20].Thisdrawbackisabsent the detailed picture of an evolution of models, are of great in the PGTG since its gravitational Lagrangian includes interest. It is well known that in GR the exact solutions of the invariants quadratic in the curvature and torsion tensors. Friedmann-Robertson-Walker (FRW) cosmological models Nevertheless the ECT is a viable theory of gravity whose constitute a basis for comparison of theoretical predictions observational predictions are in agreement with the classical with observations. On the other hand, the problem of the testsofGR,anditdifferssignificantlyfromGRonlyatvery cosmological perturbations may be correct, if it is constructed high densities of matter [17, 21, 22]. against a background of the exact solution. The ECT finds applications in cosmology [23–28], par- Other problems of the cosmology, singularities, horizons, ticle theory [19, 29, 30], and the theory of strong interac- and so forth, remain no less acute problems. The main aim of tions [31, 32]. From some time past, the interest to ECT this paper is the investigation of the possibility of the solution 2 Journal of Gravity of some problems of the modern cosmology in the frame- cosmological models with de Sitter asymptotics for work of the ECT. Furthermore, the exact integration of the the late times. ECT equations for the different combination of sources and the comparative analysis of the corresponding cosmological (ii) For <0the five types of singular models exist: models are of interest in their own right, since they allows to for the first model the scale factor () as →∞ elucidate the role of the sources of the gravitational field in grows by the law ∼ (<1); for the second cosmology. model () asymptotically tends to a finite value; other In the framework of the ECT with a nonminimally three models expand from an initial singularity, reach coupled material scalar field that has polynomial potentials themaximum,andthenbegintocontracttoafinal of the fourth degree, we here study isotropic spatially flat singularity. cosmologies in which stiff fluid is taken into account. The motivation to investigate a nonminimally coupled scalar field (iii) There are the specific values of the parameter : isbasedontheresultsobtainedwithintheframeworkof 1/6, −1/6, and −3/2. The specific feature of the value the torsionless theories of gravity in connection with the =1/6consists in the behavior of the scale factor, inflationary cosmology (see, e.g., [50–54]) and DE model when ( −min )/min ≪1: ≃min(1 + ),where 3 2 building [55–57]. On the other hand, the consideration of this ∼ for =1/6,while∼ for >1/6.The versionoftheECTincomparisonto() gravity with torsion value =−1/6is singled out as only in this case there and () gravity is motivated by the fact that, in my opinion, exists the singular expanding model, which has the the ECT requires a further study. It is relevant to remark here asymptotic |→+∞ ≃ const. From three types of the that, asit is well known, in the literature there exist a limited recollapsing models with <−1/6only for =−3/2 number of the exact cosmological solutions in the framework we have the following behavior of the scale factor: − of the ECT. |→+∞ ∼ , while for others ,weget The interest in a scalar field potential (Φ) in relativis- tic theories of gravity has been aroused by a number of 3 1 | ∼(1−√6||)/(3−√6||) − <<− , circumstances: its role in cosmology with a time variable →+∞ for 2 6 cosmological constant [58] and in quantum cosmology [59]; (1) models with (Φ) arise in alternative theories of gravity (1−√6||)/(3−√6||) 3 |→+ ∼(0 −) for <− . [60, 61] and supergravity [62]; a scalar potential governs an 0 2 inflation63 [ ] and is actively used in theories of dark matter (DM) and dark energy (DE) [7–10]. It is well known from (Φ) > GR that a period of accelerated expansion requires The exact general solutions for spatially flat FRW cos- 0 . In the paper we will consider cosmological models with mologies with a nonminimally coupled scalar field that has (Φ) > 0 (Φ) < 0 and .Thereasonstostudycosmology polynomial potentials of the fourth degree (Φ) have been with negative potentials were considered in [64]. obtained in [13]. The main results for the material scalar field A stiff cosmological fluid, with pressure equal to the are as follows. energy density, can be described by a massless scalar field, which is predicted by the string theory. On the other hand, 2 −1/2 (a) For >0, Φ <() ,and(Φ) >,the 0 thestifffluidisanimportantcomponentbecause,atearly solution describes bouncing models of four types with times, it could describe the shear dominated phase of a the accelerated expansion at late times by the laws possible initial anisotropic scenario and is dominating the ∼ ∼4/3 Φ2 =()̸ −1/2 (Φ) < 0 remaining components of the model [65]. Of no little interest and .For , , is the circumstance that this fluid leads to integrability of the the qualitative character of the model evolution does Einstein-Cartan equations. not change. In [66], the flat Friedmann universes filled by radiation, <0 Φ (Φ) > 0 stiff fluid and a nonminimally coupled ghost scalar field with (b) For ,forall , ,therearetwotypesof polynomial potentials of the fourth degree (Φ) have been collapsing models and one type of expanding models. For =−3/2there exists a unified model of DM and investigated in the framework of the ECT. Exact particular (Φ) < 0 solutionshavebeenobtainedandanalyzed.Theroleof DE.Thecasewith describes the recollapsing sourcesintheevolutionofmodelshasbeenelucidated. models. The exact general solution of an analogous problem for models containing only a nonminimally coupled scalar field The paper is organized as follows. In the next section with an arbitrary coupling constant has been obtained in we present the model and corresponding field equations. In [12]. An analysis of the solutions for the material scalar field Section 3, we obtain exact general solution of the Einstein- 2 −1/2 has shown the following. Cartan equations for >0, Φ <() .InSection 3.1 we consider the models with a scalar-torsion field and stiff fluid and discuss some interesting

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